Simplify the complex numbers: $\frac{4+i}{2-5 i}$
Answer (1)
Multiply the numerator and denominator by the conjugate of the denominator: 2 − 5 i 4 + i × 2 + 5 i 2 + 5 i .
Expand the numerator: ( 4 + i ) ( 2 + 5 i ) = 3 + 22 i .
Expand the denominator: ( 2 − 5 i ) ( 2 + 5 i ) = 29 .
Simplify the expression: 29 3 + 29 22 i .
Explanation
Understanding the Problem We are asked to simplify the complex number 2 − 5 i 4 + i . To do this, we need to get rid of the imaginary part in the denominator. We can achieve this by multiplying both the numerator and the denominator by the conjugate of the denominator.
Multiplying by the Conjugate The conjugate of the denominator 2 − 5 i is 2 + 5 i . We will multiply both the numerator and the denominator by this conjugate: 2 − 5 i 4 + i × 2 + 5 i 2 + 5 i
Expanding the Numerator Now, we expand the numerator: ( 4 + i ) ( 2 + 5 i ) = 4 ( 2 ) + 4 ( 5 i ) + i ( 2 ) + i ( 5 i ) = 8 + 20 i + 2 i + 5 i 2 Since i 2 = − 1 , we have: 8 + 20 i + 2 i − 5 = 3 + 22 i
Expanding the Denominator Next, we expand the denominator: ( 2 − 5 i ) ( 2 + 5 i ) = 2 ( 2 ) + 2 ( 5 i ) − 5 i ( 2 ) − 5 i ( 5 i ) = 4 + 10 i − 10 i − 25 i 2 Since i 2 = − 1 , we have: 4 + 25 = 29
Final Simplification Therefore, the simplified complex number is: 29 3 + 22 i = 29 3 + 29 22 i
Final Answer Thus, the simplified form of the given complex number is 29 3 + 29 22 i .
Examples
Complex numbers are used in electrical engineering to analyze AC circuits. Impedance, which is the opposition to the flow of current in an AC circuit, is a complex quantity. By simplifying complex numbers, engineers can easily calculate and analyze the behavior of these circuits, ensuring efficient and safe operation of electrical devices and systems. For example, the expression I V = Z , where V is voltage, I is current, and Z is impedance, often involves complex numbers that need to be simplified to find the real and imaginary components of impedance.
